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Full text of "Mathematical And Physical Papers - Iii"

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where M and N are arbitrary constants. If now a be small, it follows from (18) that the lateral flux at the surface, which is equal to hu, as compared with the longitudinal flux, which is equal to  K dujdx, is a small quantity of the order \f(hajK).
We may deduce the same consequence for the case of variable temperatures from the equation (17), without troubling ourselves with its solution. Conceive any number of bars of different sizes to be heated in a similar manner, and for greater generality, suppose the values of c, p, K, and h, as well as a, to be different for the different bars. Let #,#',#"... be corresponding lengths, and t, t', t" corresponding times, relating to the several bars. The equation (17) shews that the temperatures at corresponding points and at the end of corresponding times may be the same in all the bars, provided
x oc
These variations contain the definition of corresponding points and corresponding times. In order that the temperatures in the different bars should actually be the same at corresponding points and at the end of corresponding times, it is sufficient that the initial circumstances, or more generally the mode of communicating the heat, should be such as to give equal temperatures at the points and times defined by the variations (19), which in this point of view may be regarded as containing the definition of similarity of heating. Now, in comparing the longitudinal flux at corresponding points, if we take du the same, dx must vary as determined by (19), and therefore the flux will vary as K*J(K-~la~lli)} or *J(Ka~~lli), and the ratio of the lateral flux at the surface to the longitudinal flux will vary as \/(haK~~1)', so that if we suppose a to decrease indefinitely, h and K being given, the ratio in question will be a small quantity of the order tJQialK) as before, and will ultimately vanish.
The second of the variations (19) shews that if we suppose the heat to be supplied to one bar in an irregular mariner as regards the time, the fluctuations in the mode of communicating the heat must become more and more rapid as a decreases, in order that the similarity of temperatures may be kept up. If the fluctuations retain their original period, the motion of heat will tend indefinitely to become what we may regard at any instant as